1. Conceptual Issues

2. Computational Issues

Cohort-level models that track expected values.

Deterministic Models

1. Discrete Time Markov Cohort

• Characterized by transitions among mutually exclusive health states.
• Suffers from “memoryless” property, though tricks can help.
• Useful for expected outcomes, though possible to analytically solve for higher-order moments, e.g., variance, skewness; see Caswell and van Daalen (2021).

Deterministic Models

2. Differential equations modelling

• Draws on stochastic process theory.
• Processes represented by differential equations (DEQs).
• Solutions to DEQs provide occupancy of underlying process at time t.

Individual-level models that simulate patient pathways with random variation.

Stochastic Models

1. Microsimulation

   • An individual state-transition model (STM).
   • Facilitate transition probabilities that are a function of attributes. (e.g., time since disease onset, the occurrence of previous events, or time-varying response to treatment).
   • Can capture greater scope of output since the model returns the distribution of events.

Stochastic Models

2. Discrete Event Simulation

• Extends the flexibilities of microsimulation.
• Models time to event(s) explictly.
• Can incorporate interdependencies and resource contention (e.g., wait times, queueus)

Inherent errors you (generally) cannot minimize or sidestep by choosing a specific modelling approach.

Errors in Expectation

• Structural errors that result in bias.
• Attributes of models / parameters that result in model estimates that deviate from the “true” underlying event generation process.
• Not the same as heterogeneity!

Errors in Expectation

Errors in Expectation

Estimation Errors

• Sampling and estimation uncertainty.
• Focus of VOI.

Estimation Errors

Errors you can address using specific methods and approaches.

Can minimize or sidestep by choosing a specific modelling approach.

Integration Error

• Occurs when events accumulate at cycle boundaries.
• Easily corrected with 1/2 cycle correction, Simpson’s rule approaches, etc.

Integration Error

Stochastic Error

• Realisation and timing of events among otherwise identical patients may vary.
• Literature refers to this as “first-order” uncertainty.
• Can be addressed by increasing simulated patient sample M.

Stochastic Error

Embedding Error

• Standard rate-to-probability conversion formulas (i.e., P = 1 - e^{-rt}) only accurate if no competing events.
• Pairwise conversions will effectively rule out the possibility of 2 or more events in same cycle.
• Easily addressed by embedding transition probability matrix \mathbf{P} using matrix analogue to standard formula, i.e., \mathbf{P} = \exp{(\mathbf{R}t}).
• See Leech et al. (2025) and Graves et al. (2021) for techniques.

Embedding Error

Truncation Error

• Information on the precise timing of an event occurring in continuous time is lost when event times are moved to the cycle boundary point.
• Analogue: loss of power when dichotomizing a continuous variable.

Information Loss in Discrete Time

Information Loss in Discrete Time

Information Loss in Discrete Time

Truncation Error vs. Integration Error

• Integration error affects expected values.
• Truncation error affects variance.
• Addressable by increasing simulated patient size M or decreasing cycle length.

Truncation Error

What Does DES Buy You?

• No embedding error because event times are randomly sampled based on rates.
• No truncation error because exact event times used.
• DES will converge much faster than microsimulation.

What Does DES Buy You?

• Can explicitly model wait times and resource contention.
• Technical demands (roughly) on par with microsimulation.

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